random variable

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random variable

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random variable (plural random variables)

1. (statistics, loosely) A quantity whose value is random and to which a probability distribution is assigned, such as the possible outcome of a roll of a dice.
2. (statistics, formally) A measurable function from a sample space to the measurable space of possible values of the variable.
   • 1996, Ron C. Mittelhammer, Mathematical Statistics for Economics and Business, volume 78, Springer, page 45:

     Henceforth the symbol ${\displaystyle X(w)}$ will be used for the random variable ${\displaystyle X:S\rightarrow \mathbb {R} }$.

   • 2009, Christian Perwass, Geometric Algebra with Applications in Engineering, Springer, page 351:

     The particular example considered here is the Hilbert space of random variables.

   • 2012, Scott Miller, Donald Childers, Probability and Random Processes, 2nd edition, Elsevier (Academic Press), page 177:

     A two-dimensional random variable is a mapping of the points in the sample space to ordered pairs {x, y}. Usually, when dealing with a pair of random variables, the sample space naturally partitions itself so that it can be viewed as a combination of two simpler sample spaces.

Especially in discrete cases, a random variable is sometimes said to be indexed by the domain of its defining function, leading to notations such as ${\displaystyle X}$ and ${\displaystyle X_{i}}$ to represent particular values of the codomain.

• variate
• (broadly): random quantity
• (broadly, formally): aleatory variable, stochastic variable

• continuous random variable
• discrete random variable